Related papers: Weight systems and invariants of graphs and embedd…
Vassiliev invariants can be studied by studying the spaces of chord diagrams associated with singular knots. To these chord diagrams are associated the intersection graphs of the chords. We extend results of Chmutov, Duzhin and Lando to…
In a recent paper Zhuoke Yang, New approaches to ${\mathfrak gl}(N)$ weight system, Izvestiya Mathematics, 2023, vol. 77:6, 150--166; arXiv:2202.12225 (2022) a construction of a weight system, which unifies ${\mathfrak gl}(N)$ weight…
We prove that the partial-dual genus polynomial considered as a function on chord diagrams satisfies the four-term relation. Thus it is a weight system from the theory of Vassiliev knot invariants.
In real-world networks, predicting the weight (strength) of links is as crucial as predicting the existence of the links themselves. Previous studies have primarily used shallow graph features for link weight prediction, limiting the…
We associate all small subgraph counting problems with a systematic graph encoding/representation system which makes a coherent use of graphlet structures. The system can serve as a unified foundation for studying and connecting many…
A weight system on graph homology was constructed by Rozansky and Witten using a compact hyperk\"ahler manifold. A variation of this construction utilizing holomorphic vector bundles over the manifold gives a weight system on chord…
Color Lie algebras, which were introduced by Ree, are a graded extension of Lie (super)algebras by an abelian group. We show that the color Lie algebras can be used to construct universal weight systems for knot invariants of of Vassiliev…
Graph invariants provide a powerful analytical tool for investigation of abstract structures of graphs. They, combined in convenient relations, carry global and general information about a graph and its various substructures such as cycle…
This work provides the first unifying theoretical framework for node (positional) embeddings and structural graph representations, bridging methods like matrix factorization and graph neural networks. Using invariant theory, we show that…
In previous work, we defined the intersection graph of a chord diagram associated with a string link (as in the theory of finite type invariants). In this paper, we look at the case when this graph is a tree, and we show that in many cases…
We introduce a persistence-type invariant for finite weighted graphs based on combinatorial multivector dynamics. For each threshold parameter, a relation matrix determines a graph multivector field, whose induced directed dynamics admits a…
We study methods to manipulate weights in stress-graph embeddings to improve convex straight-line planar drawings of 3-connected planar graphs. Stress-graph embeddings are weighted versions of Tutte embeddings, where solving a linear system…
We define a new family of graph invariants, studying the topology of the moduli space of their geometric realizations in Euclidean spaces, using a limiting procedure reminiscent of Floer homology. Given a labeled graph $G$ on $n$ vertices…
We introduce and study so-called self-indexed graphs. These are (oriented) finite graphs endowed with a map from the set of edges to the set of vertices. Such graphs naturally arise from classical knot and link diagrams. In fact, the graphs…
The ring of graph invariants is spanned by the basic graph invariants which calculate the number of subgraphs isomorphic to a given graph in other graphs. These subgraphs counting invariants are not algebraically independent. In our view…
We study graph products of groups from the viewpoint of measured group theory. We first establish a full measure equivalence classification of graph products of countably infinite groups over finite simple graphs with no transvection and no…
Classical knot theory deals with {\em diagrams} and {\em invariants}. By means of horizontal {\em trisecants}, we construct a new theory of classical braids with invariants valued in {\em pictures}. These pictures are closely related to…
We consider the ring I_n of polynomial invariants over weighted graphs on n vertices. Our primary interest is the use of this ring to define and explore algebraic versions of isomorphism problems of graphs, such as Ulam's reconstruction…
Schreier graphs, which possess both a graph structure and a Schreier structure (an edge-labeling by the generators of a group), are objects of fundamental importance in group theory and geometry. We study the Schreier structures with which…
We develop a statistical theory to characterize correlations in weighted networks. We define the appropriate metrics quantifying correlations and show that strictly uncorrelated weighted networks do not exist due to the presence of…